On the rank of compact p-adic Lie groups

TL;DR

This paper investigates the relationship between the rank and dimension of compact p-adic Lie groups, establishing conditions under which these invariants are equal, especially for groups with trivial periodic radical and odd p.

Contribution

It proves that for certain compact p-adic Lie groups with trivial periodic radical and no (p-1)-torsion, the rank equals the dimension, extending understanding of their structural properties.

Findings

rk(G) ≥ dim(G) for all compact p-adic Lie groups

rk(G) = dim(G) when G has trivial periodic radical and no (p-1)-torsion for odd p

Provides bounds on the rank in terms of the dimension

Abstract

The rank rk(G) of a profinite group G is the supremum of d(H), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G the rank rk(G) is greater than or equal to dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical. One of our main results is the following. Let G be a compact p-adic Lie group with trivial periodic radical, and suppose that p is odd. If G has trivial (p-1)-torsion, then rk(G) = dim(G).